The advancement of biometric technology has facilitated wide applications of biometrics in law enforcement, border control, healthcare and financial identification and verification. Given the peculiarity of biometric features (e.g., unchangeability, permanence and uniqueness), the security of biometric data is a key area of research. Security and privacy are vital to enacting integrity, reliability and availability in biometric-related applications. Homomorphic encryption (HE) is concerned with data manipulation in the cryptographic domain, thus addressing the security and privacy issues faced by biometrics. This survey provides a comprehensive review of state-of-the-art HE research in the context of biometrics. Detailed analyses and discussions are conducted on various HE approaches to biometric security according to the categories of different biometric traits. Moreover, this review presents the perspective of integrating HE with other emerging technologies (e.g., machine/deep learning and blockchain) for biometric security. Finally, based on the latest development of HE in biometrics, challenges and future research directions are put forward.